Long simulations of the Solar System require integrations spanning millions of years. If the simulations are done using non-symplectic methods, the global error typically grows as t2 and the numerical solution at the end point will have few digits of accuracy. The growth of the global error can be reduced to O (t) by using symplectic methods. This reduction comes at some cost and it is not immediately obvious symplectic methods are better than non-symplectic methods. Round-off error complicates the choice. I will present numerical comparisons between non-symplectic and symplectic Runge-Kutta Nyström methods on several realistic simulations of the Solar System and discuss the trade-offs between the two types of methods.

This was joint work with R. Vaillancourt of the University of Ottawa.